As it currently stands, this question is not a good fit for our Q&A format. We expect answers to be supported by facts, references, or expertise, but this question will likely solicit debate, arguments, polling, or extended discussion. If you feel that this question can be improved and possibly reopened, visit the help center for guidance.
If this question can be reworded to fit the rules in the help center, please edit the question.

The kind of application you're looking for is called either an ATS -- Applicant Tracking System, or TMS -- Talent Management System. Put that those your search queries and you'll come up with quite a few more options.

The only ATS I have experience with from the administrative side is JobVite, but I think it would fill all your requirements. It definitely tracks opportunities, candidates, resumes, etc. It also has full workflow support -- you can set it up to move candidates through a process based on application completeness, send emails to department managers to review a new application, etc. It also supports multiple companies and divisions.

One other major player is Taleo. I've never used it as an admin, but, having been on the applying end, I'm not a fan.

Well, here’s mine. It’s called Resume Tracker. It may not be the best applicant tracking system yet, but I try really hard to deliver the best user experience and customer support. I’d love everyone’s feedback!